Multiple Residual Dense Networks for Reconfigurable Intelligent Surfaces Cascaded Channel Estimation

To greatly enhance ultra-high data rate and ubiquitous coverage requirements of the sixth-generation (6G) wireless networks, as one of the promising and innovative techniques, reconfigurable intelligent surface (RIS) aided massive multiple-input multiple-output (MIMO) is envisioned to significantly reduce link blocking probability and system energy consumption to improve link quality with sophisticated beamforming [1, 2, 3]. RIS aided MIMO has been explored with near-passive array to obtain green and sustainable communications between the user equipment (UE) and the base station (BS), by appropriately and dynamically adjusting the magnitude and phase response, wireless signals can be coherently combined and steered to desired directions [4, 5]. Each RIS reflective element can individually control the amplitude response and phase shift of the incident electromagnetic waves at the nanosecond level to achieve energy concentration. Through reflection, refraction, absorption, and transmission, the reshaped electromagnetic waves will form new paths. Based on the these passive and low-cost characteristics of RIS reflective elements, the RIS system requires very low energy consumption to improve the electromagnetic environment and increase propagation environment coverage.

However, benefits from a systematic performance improvement, the RIS system relies on the perfect channel state information (CSI) assumption. Unfortunately, the above work assumes a perfect CSI but not consider the difficulty of obtaining it. First of all, it is quite difficult to estimate the RIS-UE and RIS-BS channels separately, unless the RIS can be equipped with radio frequency (RF) chains. Secondly, the cascaded channel of the RIS between the BS and the UE can be very high-dimensional due to the massive number of reflecting elements. Currently, assuming that RIS elements are connected to RF chains, the channel estimation can be performed with acceptable performance through compressed sensing (CS) based methods. However, due to extremely low deployment, hardware and communication costs, purely passive RIS reflecting elements are undoubtedly more attractive.

By assuming active reflection patterns to achieve a smaller active array size to reduce hardware complexity, a conventional least squares method (LS) is proposed. In addition, by using the low-rank characteristics of the MIMO channel, the training overhead can be reduced through sparse matrix decomposition. Considering the sparse representation of cascaded channels, the CS method is proposed in [6]. Furthermore, as the number of antennas of the UE and BS are equipped with more antennas, the channel estimation complexity increases sharply. Using the angular-domain channel sparsity, a CS-based channel estimation scheme is proposed in [7]. However, the difference in structure sparsity between different channels will cause performance loss. Moreover, deep learning (DL) were proposed to predict the optimal RIS phase shift matrices [8], but it is still significant to get accurate CSI.

In the field of image denoising, the previous convolutional neural network (CNN) structure can construct a pair of training images by adding synthetic noise to the noise-free images [9]. Considering the similarity between image noise reduction and channel estimation, a deep residual learning approach was proposed to learn the cascaded channels from the noisy pilot-based observations [10, 11] Also, a new architecture called Multiple Residual Dense Network (MRDN) has been proposed and has received great attention [12]. In particular, the proposed architecture uses Residual Dense Network (RDN) as a component.

In this correspondence, we propose two practical residual neural networks to exploit the cascaded channel estimation. Main contributions are given as follows: First, generative adversarial networks-based convolutional blind denoising (GAN-CBD) and convolutional blind denoising network (CBDNet) are proposed to obtain accurate CSI, exploiting offline trained neural network; Second, multiple residual dense network (MRDN) is proposed to flexibly adapt to the online cascaded channel estimation; Finally, numerical results confirm that the performance of the proposed methods can significantly outperform existing schemes in terms of ADMM and CV-DnCNN, and CBDNet.

We begin by considering the uplink of a time division duplex (TDD) RIS-aided mmWave communication system. As shown in Fig. 1, we consider one RIS, one controller, one base station (BS) equipped with ${N_{b}}$ antennas and $K$ user equipments (UEs) equipped with ${N_{u}}$ antennas for the mmWave RIS-aided MIMO system, we assume that the planar RIS is equipped with $N=N_{\mathrm{v}}N_{\mathrm{h}}$ passive reflecting elements, where $N_{\mathrm{h}}$ and $N_{\mathrm{v}}$ denote the number of unit elements for RIS in horizontal and vertical orientations. Defining $\mathbf{h}_{\mathrm{r},u_{k}}\in\mathbb{C}^{{N}\times N_{u}},\mathbf{h}_{\mathrm{r},b}\in\mathbb{C}^{{N}\times N_{b}}$ as the channels from the $k$th UE to the RIS and the BS to the RIS, $\mathbf{h}_{u_{k},b}\in\mathbb{C}^{{N_{b}}\times{N_{u}}}$ as the direct channel between the $k$th UE and the BS, respectively. $\mathbb{C}^{{M}\times{N}}$ represents an ${M}\times{N}$ complex-valued matrix. Then we can express the received signal as

where $\mathbf{n}\sim\mathcal{CN}\left(\mathbf{0},\sigma_{n}^{2}\mathbf{I}\right)$ denotes the noise vector at the BS and $\sigma_{n}^{2}$ is the noise power at each antenna, $\bm{\Phi}_{k}=[\bm{\phi}_{k,1},\bm{\phi}_{k,2},\ldots,\bm{\phi}_{k,{N_{u}}}]\in\mathbb{C}^{\tau\times{N_{u}}}$ denotes pilot matrix for the $k$th UE, $\bm{\phi}_{n,k}\in\mathbb{C}^{\tau\times 1}$ denotes the orthogonal pilot sequence sent by the $n$th antenna of $k$th UE ($\bm{\phi}_{k_{1},i}^{H}\bm{\phi}_{k_{2},j}=0$, if $k_{1}\neq k_{2}$ or $i\neq j$; $\bm{\phi}_{k_{1},i}^{H}\bm{\phi}_{k_{2},j}=1$, if $k_{1}=k_{2}$ and $i=j$, $\forall k_{1},k_{2}\in\{1,2,\ldots,K\}$). For transmitting the pilots, all antennas of each UE adopt different pilot sequence. In particular, a pilot would only be allocated to one UE, resulting a orthogonal pilot matrix. Considering a simple model where one or more users in each slot have different optimal RIS phase shift matrix. Therefore, the RIS phase shift matrix $\bm{\Psi}_{k}$ represents the phase shift introduced by the RIS to the impinging signal from the transmitter in the $k$th time slot. In addition, $\bm{\Psi}_{k}\triangleq\operatorname{diag}\{\bm{\psi}_{k}\}\in\mathbb{C}^{N\times N}$, with $\bm{\psi}_{k}\in\mathbb{C}^{N\times 1}$ representing the effective phase shifts of the RIS reflecting elements and its $n$th element is $[\bm{\psi}_{k}]_{n}=\varpi_{n}e^{j\theta_{n}},\forall n\in\{1,2,\ldots,N\}$. Without loss of generality, we can assume $\bm{\psi}_{k}=\mathbf{1}$.

By exploiting $\bm{\Phi}_{k}^{\mathrm{T}}\bm{\Phi}_{k}^{*}=\mathbf{I}$ and for simplifying the designing and analysing of the channel estimation algorithms in this work, we assume that there is no direct link between UE and BS due to blockages or negligible receive power, then, the processed received signal of the $k$th UE at the BS is given by

Since practical mmWave channels usually have limited number of scatters, a LoS is expected in RIS systems. The mmWave channel of the $k$th UE to the RIS and BS to the RIS are, respectively, given as

where ${L}\ll\min\left({N_{\text{act}},N_{t}}\right)$ denotes the number of multipaths, $z_{l,k}\in\mathbb{C}$ denotes the distance-dependent pathloss of the $\mathbf{h}_{\mathrm{T},k}$ in the $l$th path. $\phi_{\mathrm{R},l}^{\mathrm{ele}}$ $(\phi_{\mathrm{R},l}^{\mathrm{azi}})$ denotes the elevation (azimuth) angle-of-arrival of the $l$th path for both $\mathbf{h}_{\mathrm{T},k}$ and $\mathbf{h}_{\mathrm{T},k_{\text{act}}}$. The steering vectors depend on the array geometry. For the typical channel ${\mathbf{h}_{\mathrm{T},k_{\text{act}}}}$ and ${\mathbf{h}_{\mathrm{T},k}}$, variables $\bm{\alpha}_{\mathrm{R}}({{\phi_{\mathrm{R},l}^{\mathrm{azi}}},{\phi_{\mathrm{R},l}^{\mathrm{ele}}}})\in\mathbb{C}^{{N_{r}}\times 1}$ and $\bm{\alpha}_{\mathrm{T}}({{\phi_{\mathrm{T},l}^{\mathrm{azi}}},{\phi_{\mathrm{T},l}^{\mathrm{ele}}}})\in\mathbb{C}^{{N_{t}}\times 1}$ are given by

where $\lambda$ denotes the wavelength, $d$ denotes the antenna spacing, and $\otimes$ is the Kronecker product. $\phi_{\mathrm{T},l}^{\mathrm{ele}}$ $(\phi_{\mathrm{T},l}^{\mathrm{azi}})$ denotes the elevation (azimuth) angle-of-departure of the $l$th path for both $\mathbf{h}_{\mathrm{T},k}$ and $\mathbf{h}_{\mathrm{T},k_{\text{ele}}}$. $\bm{\alpha}_{\mathrm{T}}({{\phi_{\mathrm{T},l}^{\mathrm{azi}}},{\phi_{\mathrm{T},l}^{\mathrm{ele}}}})$ and $\bm{\alpha}_{\mathrm{R}}({{\phi_{\mathrm{R},l}^{\mathrm{azi}}},{\phi_{\mathrm{R},l}^{\mathrm{ele}}}})$ denote the steering vectors at the sender side and the receive side, respectively.

In this section, we introduce CBDNet, GAN-CBD and MRDN for the cascaded channel estimation of RIS systems. Leveraging the sparsity of cascaded mmWave channel, we naturally introduce CBDNet-based method into cascaded channel estimation in line with previous works. And we use the GAN structure to improve the network structure. Specifically, the proposed method MRDN combines the application of residual dense network (RDN) structure and the convolutional block attention module (CBAM) [13], which serves as a building block and can obtain accurate CSI for the cascaded sparsity channel. Compared with existing baseline schemes, MRDN can reduce the complexity of RIS hardware. In the following, we will show the CBDNet, GAN-CBD and MRDN structure channel estimator. In addition, $\mathbf{x}$ and $\mathbf{z}$ represent the input and output of the universal layers and networks, respectively, in this correspondence.

We consider the RIS-aided mmWave massive MIMO system with 20 UEs, where $N_{b}=64$, $N_{u}=32$, $N=4096$, $L=3$ and $d=\lambda/2$. In terms of hardware, we use Intel Core i7-9700K @3.60GHz, 32 GB RAM and NVIDIA GeForce RTX 2080Ti to implement the above three models through PyTorch library. From the perspective of normalized mean square error (NMSE) performance, this section illustrates the pros and cons of the three proposed channel estimators in terms of structure. All simulation results are derived in PyCharm Community Edition (Python 3.8 environment). The training rate is set as 0.0001 for MRDN and 0.001 for CBDNet and GAN-CBD and the mini-batch size is 20 for all three methods. The training, validation, and testing sets include 16,000, 6,000, and 8,000 samples, respectively. The training, validation and testing sets for the three methods use the same data set samples. The number of RDN for MRDN, residual blocks for both CBDNet and GAN-CBD are 6 and 12, respectively. The MRDN has 80 features, CBDNet and GAN-CBD have 96 features. Note that NMSE is defined as

Figure 4 compares the three different models, including the MRDN, CBDNet and GAN-CBD. We can find that the MRDN can achieve best NMSE performance and fastest convergence. Because the GAN-CBD brings the advantage of judging the network, it shows better performance than the CBDNet. The computational complexity of training and offline operation can be hugely reduced. Also, the robustness of the channel estimator to different scenarios is enhanced. The average running time of MRDN (in seconds) is 0.0075, while the CBDNet and the GAN-CBD are 0.0094 and 0.0098 respectively, the computational complexity of training and offline operations for the MRDN can reduced compared with the CBDNet and the GAN-CBD. However, for almost the same computational complexity, the GAN-CBD can achieve better NMSE performance and fast convergence compared with the CBDNet. But compared with the MRDN, the improvement of network structure is not significant.

Figure 5 compares the NMSE performance of the proposed MRDN-based channel estimator for different structures (e.g., CBDNet [14], GAN-CBDN, CV-DnCNN [15]) and with existing conventional channel estimation methods (e.g., ADMM [16], PAPRFAC [17]). The simulation results are averaged over 300 iterations for the three proposed methods. It can be observed that MRDN can achieve better NMSE performance compared with GAN-CBD and CBDNet by 5.63dB and 4.51dB respectively. Compared with CV-DnCNN, which is also based on CNN, as well as conventional ADMM and PAPRFAC, regardless of the significant performance comparison in NMSE, the lower complexity of MRDN allows it to be better applied.

Figure 6 compares the NMSE performance for different number of features and RDN. With more RDNs for the global residual dense connection, and more comprehensive perceptual fields, the MRDN with 80 features and 6 dense connections for RDN performs better. Consequently, the main challenge in accurately describing noise is the lack of observational dimensions and modeling capabilities of neural networks, such as features and layers.

We proposed the CBDNet, GAN-CBD and MRDN based cascaded channel estimators for RIS-aided mmWave massive MIMO communication systems. Utilizing the sparsity of the cascaded RIS channels and classic image processing techniques, we regard the channel matrix as a two-dimensional image. The proposed residual dense network structure can increase the flexibility of the overall network to obtain better generalization and fitting capabilities, while the advantages brought by the GAN structure are not significant. Compared with the previous generation method, based on the above advantages, the MRDN-based deep learning network is designed to estimate the cascaded RIS channels. The simulation results show that the performance of the proposed the MRDN estimator increases with the increase of the scale of the network structure under the same order of complexity as the CBDNet and the GAN-CBD.